18  The Null Hypothesis

(Modified on 2026-08-11)
779 words

Recall that a hypothesis is a testable prediction which is designed to falsify a theoretical claim. An example of a hypothesis might be:

Spotify users will listen longer, on average, to familiar tracks than unfamiliar tracks.

If this hypothesis were falsified, it would call into question theories about humans’ preferences for familiar music.

In research papers, we often refer to our research hypotheses using the upper-case letter \(H\). Since we often test more than one hypothesis, it is common to number our hypotheses—\(H_1\), \(H_2\), etc. Even when we only have a single research hypothesis, we still refer to it as \(H_1\). So we might state something like:

\(H_1\): The use of rhyming words in musical lyrics increases the likelihood that listeners will understand the sung word.

Recall that refutation is easier than confirmation. It is easier to show that the statement “All swans are white” is wrong (by observing a single non-white swan) than to show that the statement is right (by observing all possible swans).

Usually, when we come up with a research hypothesis, we are hoping that it is correct. However, it is impossible to show that a hypothesis is true. It is easier, however, to show that a hypothesis is incorrect. As a result, empirical researchers restate their hypothesis in a reverse formulation. Here is an example:

This “reverse formulation” of a hypothesis is referred to as the null hypothesis. The null hypothesis is notated \(H_0\). For convenience, it is often simplified to \(H0\) without the subscript.

In empirical research, we rarely directly test the research hypothesis. Instead, we test the null hypothesis. Rather than testing the research hypothesis (and hoping that it is confirmed), we test the null hypothesis (and hope that it is rejected). Once again, we Aim not to be right, but to be not not right .

Direction vs Effect

Let’s say you hypothesize that there will be a positive relationship between tempo and spectral centroid (a timbre feature associated with “brightness”)—i.e., you predict that at faster tempos, musicians will play with a brighter tone. In mathematical terms, we expect a positive correlation between tempo and centroid: \(H_1: r > 0\), where \(r\) is the Pearson correlation coefficient between tempo and spectral centroid. What’s the null hypothesis?

You might think that the null hypothesis would be \(H_0: r < 0\)—i.e., that at faster tempos, musicians will play with a darker tone. This result, if you observed it, would definitely conflict with your hypothesis. However, there is still an effect here, even if the direction of the effect is the opposite of what we predicted. A standard null hypothesis states that there is no effect. In other words: \(H_0: r = 0\). This means that the hypothesis is “really” \(H_1: r \neq 0\). As we will learn, this approach to null hypotheses is assumed by standard statistical tests.

Of course, your true hypothesis probably does have a direction (like \(r > 0\)), and this is a good thing. To clarify that you are predicting a direction, you might state a null hypothesis like \(H_0: r \leq 0\). We’d call this a one-tailed hypothesis (we’ll learn why later). In reality, we tend to regard any effect (regardless of direction) as interesting, so we go with the more common “two-tailed” null hypothesis: \(H_0: r = 0\).

Writing about null hypotheses

In analyzing our observations, we will use statistical methods to measure not the probability that our research hypothesis is consistent with our data, but the probability that the null hypothesis is consistent with our data. If the probability is low enough, then we may conclude that the null hypothesis is inconsistent with our data. If there is reason to question the null hypothesis, our research hypothesis may be correct. However, if the null hypothesis seems like a plausible match to the data, we cannot reject it, and thus reject our research hypothesis. (It’s not research if it can’t be wrong.) We might write that “we cannot reject the null hypothesis.”

At best, we can claim that our data is not consistent with the null hypothesis.

As we know, in empirical research, we do not engage in “confirmation.” We never show things are “true.” Our data don’t “support” or “prove” or “demonstrate” anything. At best, we can claim that our data is not consistent with the null hypothesis—and conversely, that our data is consistent with the research hypothesis.

Note that many scholars historically, and still sometimes today, use unjustified language when making research claims. This could include obviously problematic statements like “we prove the research hypothesis” or “we disprove the null hypothesis.” A particularly common phrasing was “we reject the null hypothesis.” You should favor a more humble phrase like “our data is not consistent with the null hypothesis.”